%0 Journal Article
%T A Note on Average Codegrees of All Proper Subgroups of Finite Groups
%A Wei Xie
%A Shitian Liu
%J Applied Mathematics
%P 347-356
%@ 2152-7393
%D 2025
%I Scientific Research Publishing
%R 10.4236/am.2025.164018
%X Let <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>G</mi>
</math>

 be a finite group and let <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>T</mtext><mrow><mo>(</mo>
   <mi>G</mi>
  <mo>)</mo></mrow></mrow>
</math>

 be the sum of codegrees of complex irreducible characters of a group <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>G</mi>
</math>

. Say <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>f</mi><mrow><mo>(</mo>
   <mi>G</mi>
  <mo>)</mo></mrow><mo>=</mo><mfrac>
   <mrow>
    <mtext>T</mtext><mrow><mo>(</mo>
     <mi>G</mi>
    <mo>)</mo></mrow></mrow>
   <mrow>
    <mrow><mo>|</mo> <mi>G</mi> <mo>|</mo></mrow></mrow>
  </mfrac>
  </mrow>
</math>

. We prove that a finite group <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>G</mi>
</math>

 such that for all proper subgroup <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>H</mi>
</math>

, <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mn>1.1</mn><mo>&#x003C;</mo><mi>f</mi><mrow><mo>(</mo>
   <mi>H</mi>
  <mo>)</mo></mrow><mo>&#x003C;</mo><mn>1.3</mn></mrow>
</math>

, is solvable.
%K Simple Group
%K Character Codegree Sum
%K Proper Subgroup
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=141970